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Manual Reference Pages  -  MATH::LOGIC (3)

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NAME

Math::Logic - Provides pure 2, 3 or multi-value logic.

CONTENTS

SYNOPSIS



    use Math::Logic qw( $TRUE $FALSE $UNDEF $STR_TRUE $STR_FALSE $STR_UNDEF ) ;
                    #       1      0     -1     TRUE    FALSE    UNDEF

    use Math::Logic :NUM ; # $TRUE $FALSE $UNDEF -- what you normally want

    use Math::Logic :ALL ; # All the constants

    use Math::Logic :STR ; # $STR_TRUE $STR_FALSE $STR_UNDEF

    # 2-degree logic
    my $true  = Math::Logic->new( -value => $TRUE,  -degree => 2 ) ;
    my $false = Math::Logic->new( -value => $FALSE, -degree => 2 ) ;
    my $x     = Math::Logic->new_from_string( TRUE,2 ) ;

    print "true" if $true ;

    # 3-degree logic (non-propagating)
    my $true  = Math::Logic->new( -value => $TRUE,  -degree => 3 ) ;
    my $false = Math::Logic->new( -value => $FALSE, -degree => 3 ) ;
    my $undef = Math::Logic->new( -value => $UNDEF, -degree => 3 ) ;
    my $x     = Math::Logic->new_from_string( FALSE,3 ) ;

    print "true" if ( $true | $undef ) == $TRUE ;

    # 3-degree logic (propagating)
    my $true  = Math::Logic->new( -value => $TRUE,  -degree => 3, -propagate => 1 ) ;
    my $false = Math::Logic->new( -value => $FALSE, -degree => 3, -propagate => 1 ) ;
    my $undef = Math::Logic->new( -value => $UNDEF, -degree => 3, -propagate => 1 ) ;
    my $x     = Math::Logic->new_from_string( ( UNDEF, 3, -propagate ) ) ;

    print "undef" if ( $true | $undef ) == $UNDEF ;

    # multi-degree logic
    my $True   = 100 ; # Define our own true
    my $False  = $FALSE ;
    my $true   = Math::Logic->new( -value => $True,  -degree => $True ) ;
    my $very   = Math::Logic->new( -value => 67,     -degree => $True ) ;
    my $fairly = Math::Logic->new( -value => 33,     -degree => $True ) ;
    my $false  = Math::Logic->new( -value => $False, -degree => $True ) ;
    my $x      = Math::Logic->new_from_string( "25,$True" ) ;

    print "maybe" if ( $very | $fairly ) > 50 ;

    # We can have arbitrarily complex expressions; the result is a Math::Logic
    # object; all arguments must be Math::Logic objects or things which can be
    # promoted into such and must all be compatible. The outcome depends on
    # which kind of logic is being used.
    my $xor = ( $x | $y ) & ( ! ( $x & $y ) ) ;
    # This is identical to:
    my $xor = $x ^ $y ;



DESCRIPTION

Perl’s built-in logical operators, and, or, xor and not support 2-value logic. This means that they always produce a result which is either true or false. In fact perl sometimes returns 0 and sometimes returns undef for false depending on the operator and the order of the arguments. For true Perl generally returns the first value that evaluated to true which turns out to be extremely useful in practice. Given the choice Perl’s built-in logical operators are to be preferred — but when you really want pure 2-degree logic or 3-degree logic or multi-degree logic they are available through this module.

The only 2-degree logic values are 1 (TRUE) and 0 (FALSE).

The only 3-degree logic values are 1 (TRUE), 0 (FALSE) and -1 (UNDEF). Note that UNDEF is -1 not undef!

The only multi-degree logic values are 0 (FALSE)..-degree — the value of TRUE is equal to the degree, usually 100.

The -degree is the maximum value (except for 2 and 3-degree logic); i.e. logic of n-degree is n+1-value logic, e.g. 100-degree logic has 101 values, 0..100.

Although some useful constants may be exported, this is an object module and the results of logical comparisons are Math::Logic objects.

    2-degree logic

2-degree logic has one simple truth table for each logical operator.



        Perl Logic      Perl Logic     Perl Logic
    A B and  and    A B or   or    A B xor  xor
    - - ---  ---    - - --   --    - - ---  ---
    F F  F    F     F F  F    F    F F  F    F
    T T  T    T     T T  T    T    T T  F    F
    T F  F    F     T F  T    T    T F  T    T
    F T  F    F     F T  T    T    F T  T    T

      Perl Logic
    A not  not
    - ---  ---
    F  T    T
    T  F    F



In the above tables when dealing with Perl’s built-in logic T and F are any true and any false value respectively; with Math::Logic they are objects whose values are 1 and 0 respectively. Note that whilst Perl may return 0 or undef for false and any other value for true, Math::Logic returns an object whose value is either 0 (FALSE) or 1 (TRUE) only.



    my $true   = Math::Logic->new( -value => $TRUE,  -degree => 2 ) ;
    my $false  = Math::Logic->new( -value => $FALSE, -degree => 2 ) ;
  
    my $result = $true & $false ; # my $result = $true->and( $false ) ;

    print $result if $result == $FALSE ;



    3-degree logic

3-degree logic has two different truth tables for and and or; this module supports both. In the Perl column F means false or undefined; and T, F and U under Math::Logic are objects with values 1 (TRUE), 0 (FALSE) and -1 (UNDEF) respectively. The + signifies propagating nulls (UNDEFs).



        Perl  Logic        Perl  Logic         Perl  Logic
    A B and  and+ and    A B or or+  or    A B xor  xor+ xor(same)
    - - ---  ---  ---    - - -- --   --    - - ---  ---  ---
    U U  F    U    U     U U  F  U    U    U U  F    U    U
    U F  F    U    F     U F  F  U    U    U F  F    U    U
    F U  F    U    F     F U  F  U    U    F U  F    U    U
    F F  F    F    F     F F  F  F    F    F F  F    F    F
    U T  F    U    U     U T  T  U    T    U T  T    U    U
    T U  F    U    U     T U  T  U    T    T U  T    U    U
    T T  T    T    T     T T  T  T    T    T T  F    F    F
    T F  F    F    F     T F  T  T    T    T F  T    T    T
    F T  F    F    F     F T  T  T    T    F T  T    T    T

      Perl  Logic
    A not  not+ not(same)
    - ---  ---  ---
    U  T    U    U
    U  T    U    U
    F  T    T    T
    T  F    F    F

    # 3-degree logic (non-propagating)
    my $true   = Math::Logic->new( -value => $TRUE,  -degree => 3 ) ;
    my $false  = Math::Logic->new( -value => $FALSE, -degree => 3 ) ;
    my $undef  = Math::Logic->new( -value => $UNDEF, -degree => 3 ) ;

    my $result = $undef & $false ; # my $result = $undef->and( $false ) ;

    print $result if $result == $FALSE ;

    # 3-degree logic (propagating)
    my $true   = Math::Logic->new( -value => $TRUE,  -degree => 3, -propagate => 1 ) ;
    my $false  = Math::Logic->new( -value => $FALSE, -degree => 3, -propagate => 1 ) ;
    my $undef  = Math::Logic->new( -value => $UNDEF, -degree => 3, -propagate => 1 ) ;

    my $result = $undef & $false ; # my $result = $undef->and( $false ) ;

    print $result if $result == $UNDEF ;



    multi-degree logic

This is used in ‘fuzzy’ logic. Typically we set the -degree to 100 representing 100% likely, i.e. true; 0 represents 0% likely, i.e. false, and any integer in-between is a probability.

The truth tables for multi-degree logic work like this:



    and     lowest  value is the result;
    or      highest value is the result;
    xor     by truth table xor(a,b) == and(or(a,b),not(and(a,b)))
    not     degree minus the value is the result.

               Logic
     A   B  and  or xor    
    --- --- --- --- ---
      0   0   0   0   0
      0 100   0 100 100
    100   0   0 100 100
    100 100 100 100   0
      0  33   0  33  33
     33   0   0  33  33
     33 100  33 100  67
     33  33  33  33  33
    100  33  33 100  67
      0  67   0  67  67
     67   0   0  67  67
     67 100  67 100  33
     67  67  67  67  33
    100  67  67 100  33
     33  67  33  67  67
     67  33  33  67  67

     A  not 
    --- ---
      0 100
     33  67
     67  33
    100   0
  
    # multi-degree logic
    my $True   = 100 ; # Define our own TRUE and FALSE
    my $False  = $FALSE ;
    $true      = Math::Logic->new( -value => $True,  -degree => $True ) ;
    $very      = Math::Logic->new( -value => 67,     -degree => $True ) ;
    $fairly    = Math::Logic->new( -value => 33,     -degree => $True ) ;
    $false     = Math::Logic->new( -value => $False, -degree => $True ) ;

    my $result = $fairly & $very ; # my $result = $fairly->and( $very ) ;

    print $result if $result == $fairly ;



    Public methods



    new             class   object (also used for assignment)
    new_from_string class   object
    value                   object
    degree                  object
    propagate               object
    incompatible            object
    compatible              object (deprecated)
    as_string               object
    and                     object (same as &)
    or                      object (same as |)
    xor                     object (same as ^)
    not                     object (same as !)
    ""                      object (see as_string)
    0+                      object (automatically handled)
    <=>                     object (comparisons)
    &                       object (logical and)
    |                       object (logical or)
    ^                       object (logical xor)
    !                       object (logical not)



    new (class and object method)



    my $x = Math::Logic->new ;

    my $y = Math::Logic->new( -value => $FALSE, -degree => 3, -propagate => 0 );

    my $a = $x->new ;

    my $b = $y->new( -value => $TRUE ) ;



This creates new Math::Logic objects. new should never fail because it will munge any arguments into something ‘sensible’; in particular if the value is set to -1 (UNDEF) for 2 or multi-degree logic it is silently converted to 0 (FALSE). In all other cases anything that is true in Perl is converted to 1 (TRUE) and everything else to 0 (FALSE).

If used as an object method, e.g. for assignment then the settings are those of the original object unless overridden. If used as a class method with no arguments then default values are used.

-degree an integer indicating the number of possible truth values; typically set to 2, 3 or 100 (to represent percentages). Minimum value is 2.

-propagate a true/false integer indicating whether NULLs (UNDEF) should propagate; only applicable for 3-degree logic where it influences which truth table is used.

-value an integer representing the truth value. For 2-degree logic only 1 and 0 are valid (TRUE and FALSE); for 3-degree logic 1, 0, and -1 are valid (TRUE, FALSE and UNDEF); for multi-degree logic any positive integer less than or equal to the -degree is valid.

    new_from_string (class and object method)



    my $x = Math::Logic->new_from_string( 1,2 ) ;
    my $y = Math::Logic->new_from_string( TRUE,3,-propagate ) ;
    my $z = Math::Logic->new_from_string( ( FALSE, 3, -propagate ) ) ;
    my $m = Math::Logic->new_from_string( 33,100 ) ;
    my $n = Math::Logic->new_from_string( 67%,100 ) ;



This creates new Math::Logic objects. The string <B>mustB> include the first two values, which are -value and -degree respectively.

True values can be expressed as 1, T or any word beginning with T, e.g. TRUE or -true; the pattern is /^-?[tT]/. False values can be expressed as 0, F or any word beginning with F, e.g. FALSE or -false; the pattern is /^-?[fF]/. Undef values can be expressed as -1, U or any word beginning with U, e.g. UNDEF or -undef; the pattern is /^-?[uU]/. Propagate is set to true by adding a third parameter matching /^-?[tTpP1]/, e.g. -propagate. To set propagate to false either don’t include a third parameter or include it as 0 (zero).

    value (object method)



    print $x->value ;
    print $x ;



This returns the numeric value of the object. For 2-degree logic this will always be 1 or 0; for 3-degree logic the value will be 1, 0 or -1; for multi-degree logic the value will be a positive integer <= -degree.

    degree (object method)



    print $x->degree ;



This returns the degree of the object, i.e. the number of possible truth values the object may hold; it is always 2 or more.

    propagate (object method)



    print $x->propagate ;



This returns whether or not the object propagates NULLs (UNDEF). Objects using 2 or multi-degree logic always return FALSE; 3-degree logic objects may return TRUE or FALSE.

    incompatible (object method)



    print $x & $y unless $x->incompatible( $y ) ;



Returns FALSE if the objects are compatible; returns an error string if incompatible (which Perl treats as TRUE), e.g.:



    $x = Math::Logic->new_from_string(1,2) ;
    $y = Math::Logic->new_from_string(0,3) ;
    # The above are incompatible because the first uses 2-degree logic and the
    # second uses 3-degree logic.
    print $x->incompatible( $y ) if $x->incompatible( $y ) ;
    # This will print something like:
    Math::Logic(2,0) and Math::Logic(3,0) are incompatible at ./logic.t line 2102
    # The first number given is the degree and the second the propagate setting



Objects are compatible if they have the same -degree and in the case of 3-degree logic the same -propagate. Logical operators will only work on compatible objects, there is no type-coersion (but see typecasting later).

    compatible DEPRECATED (object method)



    print $x->compatible( $y ) ;



Returns TRUE or FALSE depending on whether the two objects are compatible. Objects are compatible if they have the same -degree and in the case of 3-degree logic the same -propagate. Logical operators will only work on compatible objects, there is no type-coersion (but see typecasting later).

as_string and ‘‘’’ (object method) # output: print CW$x->as_string ; # TRUE print CW$x->as_string( 1 ) ; # (TRUE,2) print CW$x->as_string( -full ) ; # (TRUE,2)



    print $x ;                      # TRUE
    print $x->value ;               # 1

    print $m ;                      # 33
    print $m->value ;               # 33
    print $m->as_string( 1 ) ;      # (33%,100)



Usually you won’t have to bother using as_string since Perl will invoke it for you as necessary; however if you want a string that can be saved, (perhaps to be read in using new_from_string later), you can pass an argument to as_string.

    and and & (object method)



    print "true" if ( $y & $z ) == $TRUE ;
    print "yes"  if $y & 1 ;
    print "yes"  if $TRUE & $y ;
   
    $r = $y & $z ; # Creates a new Math::Logic object with the resultant truth value

    print "true" if $y->and( $z ) == $TRUE ;



Applies logical and to two objects. The truth table used depends on the object’s -degree (and in the case of 3-degree logic on the -propagate). (See the truth tables above.)

    or and | (object method)



    print "true" if ( $y | $z ) == $TRUE ;
    print "yes"  if $y | 1 ;
    print "yes"  if $TRUE | $y ;
   
    $r = $y | $z ; # Creates a new Math::Logic object with the resultant truth value

    print "true" if $y->or( $z ) == $TRUE ;



Applies logical or to two objects. The truth table used depends on the object’s -degree (and in the case of 3-degree logic on the -propagate). (See the truth tables above.)

    xor and ^ (object method)



    print "true" if ( $y ^ $z ) == $TRUE ;
    print "yes"  if $y ^ 0 ;
    print "yes"  if $TRUE ^ $y ;
   
    $r = $y ^ $z ; # Creates a new Math::Logic object with the resultant truth value

    print "true" if $y->xor( $z ) == $TRUE ;



Applies logical xor to two objects. The truth table used depends on the object’s -degree. (See the truth tables above.)

    not and ! (object method)



    print "true" if ! $y == $TRUE ;
   
    $r = ! $y ; # Creates a new Math::Logic object with the resultant truth value

    print "true" if $y->not == $TRUE ;



Applies logical not to the object. The truth table used depends on the object’s -degree. (See the truth tables above.)

    comparisons and <=> (object method)

All the standard (numeric) comparison operators may be applied to Math::Logic objects, i.e. <, <=, >, =>, ==, != and <=>.

    typecasting

The only typecasting that appears to make sense is between 2 and 3-degree logic. There is no direct support for it but it can be achieved thus:



    my $x = Math::Logic->new_from_string( 1,2 ) ;  # TRUE  2-degree
    my $y = Math::Logic->new_from_string( 0,3 ) ;  # FALSE 3-degree
    my $z = Math::Logic->new_from_string( -1,3 ) ; # UNDEF 3-degree

    $x3 = $x->new( -degree => 3 ) ;
    $y2 = $y->new( -degree => 2 ) ;
    $z2 = $y->new( -degree => 2 ) ; # UNDEF converted silently to FALSE



BUGS

Multi-degree logic has a minimum degree of 4, i.e. 5-value, 0..4.

If you use & on two incompatible Math::Logic objects perl dies; I believe that this is due to a problem with overload: it does not occur with perl 5.6.0.

CHANGES

2000/05/25

No changes; just corrected an error in the tarball that meant the test would fail in some cases due to permissions problem.

2000/05/22

Dropped use of readonly pragma.

2000/04/26

Deleted quite a lot of internal error checks to improve speed.

Class is now inheritable.

2000/04/15

Have switched constants to readonly scalars, i.e. $TRUE instead of TRUE etc. This makes them easier to use for certain things, e.g. string interpolation and as array indexes or hash keys. The (now deprecated) constants still work but you are recommended to use the constant scalars instead. You will need to install readonly.pm which should be available from wherever you got Math::Logic.

The bugs with overload do not occur with perl 5.6.0. Added two tests which are run if perl’s version is > 5.005.

2000/02/27

Numerous minor documentation changes to clarify terminology.

Two bugs noted.

More tests added.

2000/02/23

Corrected multi-degree xor to match the truth table equivalence, i.e.



    xor(a,b) == and(or(a,b),not(and(a,b)))



which can be expressed in Math::Logic as



    $a->xor( $b ) == $a->or( $b )->and( $a->and( $b )->not )



or as



    $a ^ $b == ( $a | $b ) & ( ! ( $a & $b ) )



2000/02/22

Minor correction to _croak so that error messages don’t list filename and line twice; plus other minor cleanups to improve error output.

Changed the way new_from_string handles string truth values; numeric truth values operate as before.

2000/02/21

Added incompatible method and now deprecate compatible method; this provides better error messages; updated test script.

2000/02/20

Minor documentation fixes. Also eliminated a warning that occurred under 5.005.

2000/02/19

First version. Ideas taken from my Math::Logic3 and (unpublished) Math::Fuzzy; this module is intended to supercede both.

AUTHOR

Mark Summerfield. I can be contacted as <summer@perlpress.com> - please include the word ’logic’ in the subject line.

COPYRIGHT

Copyright (c) Mark Summerfield 2000. All Rights Reserved.

This module may be used/distributed/modified under the LGPL.

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perl v5.20.3 LOGIC (3) 2000-05-25

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